Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays, among the most popular numerical methods for solving partial differential equations in engineering, we encounter the finite difference and finite element methods. An alternative numerical method that has recently gained popularity for numerically solving partial differential equations is the use of artificial neural networks. Artificial neural networks, or neural networks for short, are mathematical structures with universal approximation properties. In addition, thanks to the extraordinary computational development of the last decade, neural networks have become accessible and powerful numerical methods for engineers and researchers. For example, imaging and language processing are applications of neural networks today that show sublime performance inconceivable years ago. This dissertation contributes to the numerical solution of partial differential equations using neural networks with the following two-fold objective: investigate the behavior of neural networks as approximators of solutions of partial differential equations and propose neural-network-based methods for frameworks that are hardly addressable via traditional numerical methods. As novel neural-network-based proposals, we first present a method inspired by the finite element method when applying mesh refinements to solve parametric problems. Secondly, we propose a general residual minimization scheme based on a generalized version of the Ritz method. Finally, we develop a memory-based strategy to overcome a usual numerical integration limitation when using neural networks to solve partial differential equations.

翻译：偏微分方程在物理、生物或社会现象的建模中具有广泛应用，因此我们需要以计算可行的方式近似求解这些方程。当前，工程中求解偏微分方程最常用的数值方法包括有限差分法与有限元法。近年来，一种新兴的偏微分方程数值求解方法——人工神经网络——逐渐受到关注。人工神经网络（简称神经网络）是具有通用逼近特性的数学结构。此外，得益于过去十年计算技术的非凡发展，神经网络已成为工程师和研究者可获取且强大的数值工具。例如，图像处理与语言处理如今是神经网络的应用领域，其表现之卓越在数年前尚难以想象。本论文致力于利用神经网络数值求解偏微分方程，目标双重要求：探究神经网络作为偏微分方程解逼近器的行为特性，并提出基于神经网络的方法用于传统数值方法难以处理的框架。作为新型神经网络方法，我们首先提出一种受有限元法启发、通过网格细化求解参数化问题的方案。其次，我们基于Ritz方法的广义形式提出通用残差最小化框架。最后，我们发展了一种基于记忆的策略，以克服使用神经网络求解偏微分方程时常见的数值积分局限性。